Dihedral Hamiltonian cycle systems of the cocktail party graph

The existence problem for a Hamiltonian cycle decomposition of $K_{2n} − I$ (the so called cocktail party graph) with a dihedral automorphism group acting sharply transitively on the vertices is completely solved. Such Hamiltonian cycle decompositions exist for all even n while, for n odd, they exist if and only if the following conditions hold: (i) n is not a prime power; (ii) there is a suitable $\ell$ such that p ≡ 1 (mod $2^\ell$) for all prime factors p of n and the number of the prime factors (counted with their respective multiplicities) such that $p \not\equiv 1$ (mod $2^{\ell+1}$) is even. Thus in particular one has a dihedral Hamiltonian cycle decomposition of the cocktail party graph on 8k vertices for all k, while it is known that no such cyclic Hamiltonian cycle decomposition exists. Finally, this paper touches on a recently introduced symmetry requirement by proving that there exists a dihedral and symmetric Hamiltonian cycle system of $K_{2n}-I$ if and only if n≡2 (mod 4).